\magnification = 2200\hsize 7 true in\vsize 9 true in\hoffset = -0.20 true in\voffset -0.25 true in\parskip=3pt\input amssym.def            % small letters for UNIX,  not: AMSsym.def\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'\def\cl{\centerline}\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum\nopagenumbers\vglue 10pt\lf%%%%%%%%%%%%%%%%%%%% Plain TEX\def\LF{\medskip\noindent}\font\cmrX=cmbx10 scaled \magstep 1\centerline     {\cmrX Surfaces of revolution with}
\centerline     {\cmrX constant Gau\ss\ curvature}\LFA surface of revolution is usually described by giving its meridiancurve $s\mapsto(r(s), h(s))$. The surface is then obtained by rotation:$$(x,y,z) := (r\cos\varphi, r\sin\varphi, h).$$ 
Any kind of curvature conditioncan be expressed as a differential equation for the meridian curve.\LFThe case of constant Gau\ss\ curvature $K$ is particularly simple if themeridian is parametrized by arclength, i.e., $r'^2+h'^2=1$. In this casethe meridian is determined by$$ r''(s) + K\cdot r(s) = 0, \hskip5mm h(s) = \int_0^s \sqrt{1 - r'(t)^2}dt.$$We describe the three kinds of examples in the case $K=1$.
\vfil\eject
\hbox{ Sphere:}
\vskip -12pt$$ r(s) = \sin(s),\hskip5mm 0\le s \le\pi $$
\vskip -2pt
\hbox{ With cone points:} 
\vskip -14pt$$r(s) =a\sin(s),\ \ 0\le s \le\pi ,\ 0<a<1 $$
  \hbox{ With singularity curve:}$$r(s) =a\sin(s),\ \ b\le s \le\pi-b ,\ 1< a, \  \cos(b):=1/a.$$  \bye\noindentM.W.
\bye